Theorem REF  has also been generalized in numerous ways: Belabas–Bhargava–Pomerance [1]} prove power saving error terms; Bhargava [2]}, [3]}
determines the asymptotics of quartic and quintic fields, when
ordered by discriminant; Taniguchi [4]} and Bhargava–AS–Wang [5]} count cubic extensions of number fields and function fields; Belabas–Fouvry [6]} count subfamilies of cubic fields satisfying congruence conditions on their discriminants; Terr [7]} proves that the “shapes” of cubic rings and fields are equidistributed (see also work of Bhargava–Harron [8]}, who give a uniform proof that shapes of cubic, quartic, and quintic rings and fields are equidistributed); Taniguchi–FT [9]} and Bhargava–AS–Tsimerman [10]} compute secondary terms (of size \(\asymp X^{5/6}\) ) for the asymptotics of \(N^{\pm }_{\operatorname{Disc}}(X)\) .
